We study[1, 2] electronic transport through a Luttinger liquid with an embedded weak scatterer (WS) or weak link (WL) in the presence of the electron coupling to one-dimensional massless bosons (phonons). We have found that for an arbitrary fixed strength of boson scattering by the impurity the scaling dimensions of the conductance in the WS and WL cases obeys the duality condition, ∆WS∆WL = 1, established for the standard Luttinger liquid [3] (i.e. in the absence of the additional coupling). On the other hand, with the strength of both fermion and boson scattering by impurity changing from the WS to the WL limit, the system has a rich phase diagram with a metal-insulator transition as a function of a bare backscattering amplitude. The transition exists in a certain interval, defined by the relative strength of the electron-electron and electron-boson coupling and the ratio of Fermi to sound velocities.

[1] A. Galda, I. V. Yurkevich, and I. V. Lerner, Phys. Rev. B 83, 041106 (2011); EPL 93, 17009 (2011). 
[2] I. V. Yurkevich, A. Galda, O. M. Yevtushenko, and I. V. Lerner, to be published (2012). 
[3] C. L. Kane and M. P. A. Fisher, Phys. Rev. Lett. 68, 1220 (1992); Phys. Rev. B 46, 15233–15262 (1992).